Interesting puzzle that is quite hard to stop.
4 moves, as far as I can tell.
And one more:
Same theme for all of them, really.
I think it is 5 moves.
You can't do it any shorter as you need at least 4 moves to get your pawn to the 2nd line where it can checkmate the white's king and one more move to bring a supporting piece (queen or bishop), so the white king wouldn't be able to capture your pawn.